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# Questions tagged [arithmetic-geometry]

Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, Mochizuki theory.

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### Is a birational morphism between normal projective varieties residually separable?

My goal is to use a "normal Bertini" theorem (see https://link.springer.com/article/10.1007%2Fs000130050213) More specifically, let $k$ be a field (you may assume that k is infinite but it should be ...
230 views

### Can someone help in how to approach reading Mordell-Weil Theorem for abelian varieties?

I was thinking to start reading the proof of Mordell-Weil Theorem for abelian varieties over number fields, after getting done with the proof in the case of elliptic curves over number field and I ...
814 views

### Bounded Torsion, without Mazur’s Theorem

Mazur’s torsion theorem famously tells us exactly which finite groups can occur as the torsion subgroup of $E(\mathbb{Q})$ for an elliptic curve $E$ defined over $\mathbb{Q}$. In particular, it ...
50 views

This is from J.S.Milne's Elliptic Curves book. We have $H^1(\mathbb{Q}, E_2) \cong (\mathbb{Q}^× / \mathbb{Q}^{×2})^2$ because $Gal(\mathbb{Q}^{al} / \mathbb{Q})$ acts trivially on $E_2(\mathbb{Q}^{... 0answers 65 views ### Points on an Elliptic Curve, how to interpret$(x(2P):z(2P))$? [on hold] This is from J.S. Milne's book 'Elliptic Curves'- With$E(\mathbb{Q}) : Y^2Z = X^3 +aX Z^2+bZ^3$and given a point on$P =(x,y)$on it's Weierstrass equation dehomogenized$E: y^2= x^3+ax+b$where$a,...
101 views

### Having difficulty in understanding a result that'll help in proving the finiteness of Selmer group

I'm reading and trying to understand the proof of the finiteness of n-Selmer group from J.S.Milne's Elliptic Curves book but having difficulty in understanding it. Here's a screenshot from the book- ...
272 views

### Fontaine-Fargues curve and period rings and untilt

When I read the paper "THE FARGUES–FONTAINE CURVE AND DIAMONDS" of Matthew Morrow, I have a question on page 11. Question: The arthur said that the de Rham and crystalline period rings implicitly ...
239 views

### How to prove the p-adic Galois representations atteched to the Tate module of an abelian variety is de Rham directly?

Recently I read a thesis p-adic Galois representations and elliptic curves. Using Tate's curve, the author proved the p-adic Galois representations atteched to the Tate module of an elliptic curve is ...
528 views

### Classify 2-dim p-adic galois representations

Recently I have known how to classify 1-dim p adic Galois representations $\phi$. The p-adic Galois representations mean that a representation $G_K$ on a p-adic field $E$, where $K$ is also a p-adic ...
174 views

### Complex isomorphism class of abelian varieties and $L$-functions

In his famous Mordell paper, Faltings proved that two abelian varietes $A_1, A_2$ defined over a number field $K$ are isogenous if and only if the local $L$-factors of $A_1, A_2$ are equal at every ...
199 views

### Can someone help clear up some confusion regarding the proof of Mordell-Weil theorem?

I'm reading the proof(s) of Mordell-Weil theorem using various texts. This post is to make sure what I'm reading and understanding is correct.. Rational points on Elliptic Curves by Silverman Tate ...
107 views

### Dimension of Prym variety of cover

I am reading the article by Lawrence and Venkatesh on diophantine problems and $p-$adic period mappings. At page $35$ they say that the dimension of the Prym variety of an (unramified) cover of curves ...
68 views

### Tate modules of Jacobian varieties isomorphic over $\overline{\mathbb{Q}}$ but not over a number field $K$

Let $C_1, C_2$ be two curves defined over a number field $K$. Suppose that $C_1, C_2$ are isomorphic over $\overline{\mathbb{Q}}$ but not over $K$ and that $C_1(K), C_2(K) \ne \emptyset$. Then the ...
394 views

### Is every positive integer the rank of an elliptic curve over some number field?

For every positive integer $n$, is there some number field $K$ and elliptic curve $E/K$ such that $E(K)$ has rank $n$? It's easy to show that the set of such $n$ is unbounded. But can one show that ...
106 views

### Which Shimura varieties admit or don't admit $p$-adic uniformization by Drinfeld spaces?

$p$-adic uniformization is a powerful tool for studying Shimura curves and Shimura varieties. For instance, we know cohomology groups of Drinfeld spaces, so we know some information about the Shimura ...

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